The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth which can be exponential in the worst case, as for the Vietoris–Rips. In this talk I will recast this problem at the level of covers, developing a framework in which filtrations and persistence modules can be constructed, analyzed, and compared through simple relations between covers rather than at the level of simplicial complexes. The guarantees propagate through any functor that preserves the contiguity of refinement maps, we give the example of two such functors: the Nerve and the Co-Nerve. I will show that working at this level is drastically simpler, with stronger, more general consequences. I'll then explore this perspective and show how it can be used to construct a robust approximation of the Vietoris-Rips filtration that is orders of magnitude smaller, while maintaining a \( \log 3\)-interleaving unconditionally for any metric space.
 

References:

• Leitão, A. (2026). It's All About Covers: Persistent Homology of Cover Refinements. arXiv preprint arXiv:2602.22784.
• Virk, Ž. (2021). Rips complexes as nerves and a functorial Dowker-nerve diagram. Mediterranean Journal of Mathematics, 18(2), 58.
• De Silva, V., Munch, E., & Stefanou, A. (2017). Theory of interleavings on categories with a flow. arXiv preprint arXiv:1706.04095.
• Kerber, M., & Schreiber, H. (2019). Barcodes of towers and a streaming algorithm for persistent homology. Discrete & computational geometry, 61(4), 852-879.